A numerical semigroup is said to be perfect if it does not contain any isolated gaps. In this paper, we will look at some basic properties of isolated gaps in numerical semigroups. In particular, we will see how they are related to elements of the Apery set. We will use these properties to find all of the isolated gaps in a numerical semigroup of embedding dimension two and demonstrate a simple method of generating some examples of perfect numerical semigroups of embedding dimension three.
Numerical semigroup, frobenius number, gap, chinese remainder theorem
SMITH, HAROLD J.
"On isolated gaps in numerical semigroups,"
Turkish Journal of Mathematics: Vol. 46:
1, Article 9.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss1/9